The frog model with death and drift on free products of complete graphs

We study the frog model with death and drift on $\mathbb{D}_{m,d}$, the free product of $d+1$ copies of the complete graph of order $m$. Active and inactive particles are located at the vertices of $\mathbb{D}_{m,d}$. Each active particle performs a $\alpha$-biased random walk towards the root of $\mathbb{D}_{m,d}$, dying after a random lifetime with a geometric distribution of parameter $1-p$. Each inactive particle remains dormant until an active particle visits its location. We present conditions on the parameters $\alpha$ and $p$ for the process to die out almost surely and to survive with positive probability. Our proofs are based on comparisons of the model with simple and multi-type branching processes.